1. **State the problem:** We have a large triangle on a straight line base with angles 46°, 63°, y, 140°, and x as shown. We need to find the values of $x$ and $y$. 2. **Identify known angle relationships:** The straight line base means angles on the line sum to 180°. 3. **Use the straight line angle sum:** The angle adjacent to 140° on the straight line is $180° - 140° = 40°$. This angle is $x$ because it is the angle at the base where the interior segment meets the line. 4. **Use triangle angle sum rule:** The top vertex angle is split into two parts: 63° and $y$. The sum of angles in the triangle is 180°, so for the triangle with angles 46°, 63° + $y$, and the angle at the base where the segment meets the line (which is $x$), we have: $$46° + (63° + y) + x = 180°$$ 5. **Substitute $x = 40°$ from step 3:** $$46° + 63° + y + 40° = 180°$$ 6. **Simplify and solve for $y$:** $$149° + y = 180°$$ $$y = 180° - 149° = 31°$$ **Final answers:** $$x = 40°$$ $$y = 31°$$